JP2019064541A - Ducted fan, multi-copter, vertical takeoff and landing aircraft, cpu cooling fan and radiator cooling fan - Google Patents

Abstract

To provide a ducted fan that can maximize static thrust.SOLUTION: A ducted fan includes a duct 10, a propeller 20, a motor 30, a housing 40 and a stator 50. The propeller 20 includes a hub 21 disposed concentrically with the duct 10; and four blades 22 provided to an outer periphery of the hub 21 at equal intervals. The blade 22 has a chord length that gradually toward a wing tip from a root, while the blades 22 has a chord length that extends to the blade tip from the vicinity part of the blade tip.SELECTED DRAWING: Figure 1

Description

  The present invention relates to a ducted fan used to obtain a propulsive force and cooling air, a multi-copter and a vertical take-off and landing machine equipped with the ducted fan, and a CPU cooling fan and a radiator cooling fan including the ducted fan.

  The multicopter type drone extends not only observation and photography but also to the transportation of goods. The multicopter is becoming larger in size to carry larger and heavier supplies, and there is concern about personal injury due to blades rotating with large power. There are cases where a guard is attached to the outside of the propeller to improve personal safety. Further, in ETOP (Patent Document 1) of Israel, a ducted fan is used. In manned aircraft, VZ-8 manufactured by Piasecki of the United States (Patent Document 2) and Cormorant of Israel (Patent Document 3) use two ducted fans.

  The advantages of using a ducted fan are not only high personal safety but also noise reduction by attaching a sound absorbing material to the inner surface of the duct, the lip of the duct generates thrust, and the same power and conventional propeller as long as it has the same diameter. In theory, it is possible to give a thrust of 1.27 times (Non-Patent Document 1), and 1.4 times in experiments.

U.S. Patent Publication No. 2013-0233964 U.S. Pat. No. 3,184,183 U.S. Patent Publication No. 2011-0049307

"Airplane Design Theory" Masao Yamana, Hiroshi Nakaguchi Yokendo 1980, pp. 388-393. Betz, A., Prandtl, L .: Schraubenpropeller mit geringstem Energieverlust (Screw Propellers with Minimum Induced Loss), Goettingen Reports 1919, pp. 193-213. Harada Masashi: Rotor design method for maximizing static thrust, Transactions of the Japan Aerospace Society, August, 2017.

Non-Patent Document 2 derived a method for determining the optimal shape of a low disk load propeller blade. The problem of maximizing the thrust of a stationary rotor has been reported by the inventor (3). However, this report is applicable only when the disc load is small as at the time of cruising, and can not be applied to a ducted fan which maximizes the stationary thrust.
An object of the present invention is to provide a ducted fan capable of maximizing stationary thrust.
An object of the present invention is to provide a multi-copter and a vertical take-off and landing aircraft equipped with such a ducted fan.
An object of the present invention is to provide a CPU cooling fan and a radiator cooling fan comprising such a ducted fan.

  In order to achieve the above object, the ducted fan according to one aspect of the present invention has a duct having an R-shaped lip at an outlet, a blade tip close to an inner wall of the duct, and a chord length approaching the blade tip The propeller has a blade which gradually decreases and extends from the vicinity of the blade tip to the blade tip or has a fixed length.

  According to the inventor's knowledge, since the blade tip of the blade and the inner wall of the duct are in close proximity, the vortex flow through the gap between them is substantially eliminated and the energy loss due to the vortex flow is reduced. In addition, by gradually reducing the chord length of the blade as it approaches the wing tip, it is possible to make the flow in the duct close to uniform flow while securing the strength. Furthermore, the blade chord length increases from near the blade tip to the blade tip, increasing the flow velocity outside the duct and increasing the thrust due to the upward suction force generated by the flow velocity around the R-shaped lip of the duct Do. Gain due to energy loss reduction due to flow equalization in the duct due to tapering towards a moderate blade chord length wingtip, and thrust increase due to increased velocity inside and outside the duct due to blade tip enlargement The gain is an optimal value without losing both, and the stationary thrust can be maximized.

  According to the present invention, the stationary thrust of the ducted fan can be maximized.

It is a top view showing a ducted fan concerning one embodiment of the present invention. It is AA sectional drawing of FIG. It is an enlarged view of the area | region shown by code | symbol B of FIG. It is a graph which shows an example of distribution of the chord length of the blade in the ducted fan concerning one embodiment of the present invention. It is a graph which shows the exit speed of the airflow in the outlet of the duct in the example concerning the present invention. It is a graph which shows the blade angle of the blade in the Example which concerns on this invention. It is a graph which shows the Reynolds number in the Example which concerns on this invention. It is a figure which shows the leading edge thrust which acts on a flat blade and a duct. It is a figure which shows the duct model with a flange. It is a figure which shows the moving speed of the vortex which exists in the boundary of a wall surface and uniform flow. It is a figure which shows the hierarchical structure of a vortex lattice. It is a figure which shows the coordinate system of a ducted fan. It is a figure which shows a braid | blade, the discharge | release vortex, and a wing tip vortex. FIG. 7 shows discretization of blades and emitting vortices. It is a figure which shows the flow which flows in into a braid | blade. It is a figure which shows the force which acts on a braid | blade. It is a figure which shows the detail of a duct trailing edge. FIG. 2 illustrates a ducted fan with a hub. It is a graph which shows the comparison of blade shape. It is a graph which shows comparison (with a hub) of blade shape. It is a graph which shows comparison (with a hub) of vortex surface movement speed. It is a graph which shows thrust and a lip factor.

Hereinafter, embodiments of the present invention will be described with reference to the drawings.
FIG. 1 is a top view showing a ducted fan according to an embodiment of the present invention, and FIG. 2 is a sectional view taken along the line A-A of FIG.
The ducted fan 1 has a duct 10, a propeller 20, a motor 30, a housing 40, and a stator 50.

  The duct 10 has a cylindrical duct body 11 and a flange 13 provided on the outlet 12 side of the duct body 11. The outlet 12 of the duct body 11 has an R-shaped lip 14.

The propeller 20 has a hub 21 arranged concentrically with the duct 10 and four blades 22 provided at equal intervals on the outer periphery of the hub 21. The number of blades 22 is typically four, but two to eight are preferable. Further, the number of blades 22 is, for example, four when the number of stators 50 is seven, so that the least common multiple of the number of stators 50 and the number of stators 50 has a large value. And the position of the stator 50 can be avoided at the same time, and noise due to interference can be reduced.
The hub 21 is rotationally driven by a motor 30 housed in a housing 40. Thereby, the blade 22 rotates.

The plurality of stators 50 are fixedly disposed so as to bridge between the outer periphery of the housing 40 and the inner wall 15 of the duct 10. Thus, the housing 40 is fixedly disposed at the center of the duct 10.
FIG. 3 is an enlarged view of a region indicated by reference symbol B in FIG.

The blade 22 has its wing tip 22A close to the inner wall 15 of the duct 10. It is preferable that the gap G between the wing tip 22A and the inner wall 15 of the duct 10 be as small as possible without the wing tip 22A coming into contact with the inner wall 15 of the duct 10.
FIG. 4 is a graph showing an example of the distribution of the chord length CL of the blade 22. As shown in FIG.

  The chord length CL of the blade 22 gradually decreases from the root toward the tip 22A. That is, the chord length CL of the blade 22 is the largest at the root and gradually decreases toward the tip. However, the chord length CL of the blade 22 is expanded from the tip vicinity portion 22B to the tip 21A. The chord length CL may be a constant length from the tip end vicinity portion 22B to the tip end 22A.

The chord length CL of the blade 22 gradually decreases as it approaches the blade tip 22A from the root, so that the flow in the duct can be made close to uniform flow, and the strength of the blade 22 can be easily secured, and the aerodynamic advantage is excellent. Thin airfoils can be used from root to tip.
Here, when the chord length of the wing tip 22A of the blade 22 is CL _A , and the chord length of the wing tip vicinity portion 22B of the blade 22 is CL _B ,
1.0 × CL _B ≦ CL _A ≦ 1.3 × CL _B
Is preferred. When CL _A exceeds 1.3 × CL _B , the gas flow in the duct 10 is rather uneven.

Thus, the position of the tip end vicinity portion 22B where the chord length CL of the blade 22 is expanded is within 15% of the radius (r in FIG. 4) of the blade 22 from the tip 21A (r _{15 in} FIG. 4). Is preferred. If the position of the tip vicinity portion 22B exceeds 15% of the radius of the blade 22, that is, if the chord length CL decreases as it approaches the tip 22A too much, the duct decreases due to the tapering of the chord length CL of the blade 22 The flow uniformity within 10 becomes insufficient.
It is preferable that the degree of the gradual reduction of the chord length CL of the blade 22 becomes smaller as it approaches the tip 22A. Due to the chord length distribution where this tapering is reduced, the flow in the duct approaches uniformly and induced energy losses are minimized.

The ratio of the maximum chord length of blade 22 (the chord length CL _C of the root of blade 22) and the minimum chord length (the chord length CL _B of tip vicinity portion 22B) is 3.0 or more at 1.5 or more It is preferable that it is the following. When designed to generate a large thrust by limiting the blade tip speed, this taper ratio decreases and approaches 1.5 and approaches 3.0 when a small thrust is generated. Therefore, the optimum taper ratio is between 1.5 and 3.
The diameter Dh of the hub 21 is preferably 10% or more and 50% or less of the inner diameter Rd of the duct 10. If it is less than 10%, the chord length of the blade root is designed to be small, and it becomes difficult to maintain the strength. If it is more than 50%, the area of the hub occupying the area of the blade rotation surface becomes too large, and the thrust that can be generated decreases.
When the lip 14 of the discharge port 12 of the duct body 11 has an inner diameter of the duct 10 as Rd,
3/100 × Rd ≦ r _r ≦ 30/100 × Rd
It is preferable to have an R-shape with a radius of curvature r _r of If it is smaller than 3/100, the flow around the lip peels off. If it is larger than 30/100, the lip will be larger than necessary, leading to a meaningless enlargement of the ducted fan.
The length Ld (see FIG. 2) of the duct 10 is preferably longer than 25% of the inner diameter Rd of the duct 10. If it is less than 25%, contraction of the flow occurs behind the duct and the figure of merit is reduced.
(Example)
The Example of the ducted fan 1 comprised in this way is shown.
Table 1 shows the design conditions, and Table 2 shows the performance of the ducted fan 1 designed.

表２に示すように、推力は６３．２Ｎ、フィギュアオブメリットは９１．８に達した。

As shown in Table 2, the thrust reached 63.2 N and the figure of merit reached 91.8.

  The chord length distribution of the ducted fan 1 of this embodiment is as shown in FIG. The chord length is the largest at the point A of the root of the blade 22 and is 4.77 cm. The chord length decreases from point A to point B, the rate of decrease is the largest at point A, the rate of decrease decreases toward the tip, and becomes a minimum point at point B. The radial position of point B is 11.7 cm, which is 85.2% of the blade radius. Also, the chord length at point B is 1.71 cm, which is 35.9% of the value at point A. The chord length increases from point B to tip 22A and reaches point C at the tip. The chord length at point C of the wing tip is 2.06 cm, 1.2 times the value at point B.

FIG. 5 shows the outlet velocity v of the air flow at the outlet 12 of the duct 10 in this embodiment.
As shown in FIG. 5, the outlet velocity v monotonously increases from the point D at the root of the blade 22 to the point E at the blade tip 22A, and can be approximated by a linear function. This linear function is represented by the radius of the blade 22 as r: v = ar + b
Is approximated by the following equation, the blade angle θ is θ = tan ⁻¹ ((ar + b) / rΩ) + θ ₀

Given by Here, Ω is the blade rotational angular velocity, and θ ₀ is, for example, the attack angle of the blade (blade 22) giving a lift coefficient of 0.6. The blade angle of such a blade 22 is shown in FIG.
FIG. 7 shows the Reynolds number in this embodiment.

  As shown in FIG. 7, the Reynolds number is maintained at a substantially constant value over the wide radius of the blade 22, and there is no region of low Reynolds number. Therefore, it is not susceptible to the increase in resistance due to the low Reynolds number. Therefore, high figure of merit can be obtained even with a small ducted fan.

(Proof)
As described above, the present invention can maximize the static thrust of the ducted fan. The following proves this.
1. Problems in Design for Maximizing Static Thrust 1-1. The problem of leading edge thrust occurring at the lip In a flat blade, as shown in FIG. 8 (a), the flow around the leading edge of the wing causes a large negative pressure at the leading edge, and this negative pressure causes the force in the thrust direction to be at the leading edge. work. In Vortex Lattice Method (hereinafter, VLM), since the blade is approximated by a flat plate, the leading edge area is 0, and the flow velocity is infinite. Therefore, it is necessary to use a special calculation method to calculate this leading edge thrust (see Weissinger, J .: The Lift Distribution of Swept-Back Wings, NACA TM-1120, 1947.). However, this method is applicable only when the potential of the target surface is continuous.

Similarly, as shown in FIG. 8B, when the disc load is large (when the advancing speed V is not sufficiently larger than the axial induction speed v), there is a flow around the lip, and the leading edge thrust acts on the lip. Since the tip discharge vortices are spirally attached to the rear of the fan surface inside the duct, the potential becomes discontinuous in the flow direction, and VLM can not calculate this leading edge thrust.
In the non-patent document 3, in order to avoid the calculation of the leading edge thrust, the inventor performed the calculation under the condition that the flow around the lip can be neglected, that is, V> v.

Here the leading edge thrust ducted fan stationary in question is large, of the total thrust T according to the simple momentum theory, edge thrust T _D before acting on the lip reaches 50% of the total thrust T (see Non-Patent Document 1 ). Therefore, it was not possible to determine the exact total thrust T of the ducted fan using VLM.
1-2. Problem of spiral vortex pitch

In the conventional propeller theory, the pitch of the spiral vortices emitted from the blade is determined from the blade rotational angular velocities Ω and V, assuming that the induction velocity v is negligibly small with respect to the forward velocity V. However, in the problem of maximizing thrust at rest, this forward velocity V is zero. Therefore, the conventional method of determining the pitch of a spiral vortex can not be used.
1-3. Problem of momentum mismatch in the Trefftz plane

The total thrust T of the ducted fan must be equal to the thrust T _T (hereinafter referred to as Trefftz surface thrust) calculated from the velocity distribution on the Trefftz surface. Initially, the total thrust T and the Trefftz thrust T _T did not match.
1-4. VLM accuracy issues

This problem is not limited to the problem of the ducted fan design which maximizes the static thrust precisely, but also occurs in the design of the ducted propeller when the disc load is small. Because the tip emitting vortices flow along the duct surface, the change in the circulation of the vortex lattice around the tip emitting vortices is large. So, if you break up the grid to catch this change, the computation time will be huge.
2. How to solve the problem

The problem of the leading edge thrust of 1-1 can be avoided by flanged the duct and providing a curvature at the connection with the duct as shown in FIG. Hereinafter, a portion having this curvature is called a lip. The flow around the tip of the flange generates a leading edge thrust here, but if the radius of the flange is large enough, the flow velocity at the flange tip becomes sufficiently small and the leading edge thrust becomes negligibly small. Furthermore, the leading edge thrusts cancel each other because the leading edge thrusts having the same magnitude and opposite direction are generated on the opposite side of the flange. The leading edge thrust at the lip produced by the straight duct without flange should be integrated on the lip and flange surface with the axial component of the negative pressure created by the flow around the lip of the flanged duct in FIG. 9 and the flange. It is obtained by It is shown in the calculation described later that this leading edge thrust depends on the curvature radius R _R of the lip.

The problem 1-2 was solved as follows. First, the induction velocity obtained from the simple momentum theory is set as the initial vortex surface movement velocity v ₀ ′, and optimization calculation is performed. Next, the vortex surface movement velocity v 'given by the following equation is calculated from the axial induced velocity w (a negative value is taken in the coordinate system described later) and the induced velocity v in the rotational direction of the blade obtained.

ここでφは次式で与えられる

Where φ is given by

  Since the vortex surface movement velocity v 'is not constant in the radial direction, curve fitting is performed using a simple function, and the pitch is determined using that function. Optimal design is performed using this new spiral vortex, and a new spiral vortex is generated. The update of the spiral vortex is repeated, and it is determined that the solution has converged at a stage below the truncation error.

Problems 1-3 were solved by considering that the duct surface and the duct discharge vortex described later are at the stationary duct wall surface or the boundary with the atmosphere. According to the theory of superfunctions (Isao Imai: Applied Superfunction Theory I, Science, 1981.), when there is a flow of velocity V along a stationary wall as shown in FIG. The vortex is moving at a velocity of V / 2. Similarly, the movement speed of the tip discharge vortex on the duct surface, and the movement velocity of the duct trailing edge horseshoe vortex (described later) at the boundary between the static atmosphere and the flow in the duct are all the thrust by setting the vortex surface movement velocity v 'just inside. T and Trefftz surface thrust T _T coincide with each other.

In the problem 1-4, the duct surface is not divided by a grid of a certain size as shown in FIG. 11, but a grid having a hierarchical structure as shown in FIG. 11 is used. The size of the grid near the tip vortices should be (1/2) ⁿ times that of the reference grid. FIG. 11 shows the case of n = 2. By using a triangle for the grid and taking the hypotenuse to coincide with the helix angle of the tip vortex, it is possible to place a fine grid around the tip vortex where the change in circulation distribution is large.
3. Optimization Method 3-1. Ducted fan model

The coordinate system of the ducted fan is taken as shown in FIG. The grids on the surface of the duct (hereinafter referred to as duct grids) are given an arbitrary order, and the circulation of the ith duct grid is denoted by 図_Di as shown in FIG. Also, control points are provided at the center of gravity of the grid. A spiral horseshoe vortex (hereinafter referred to as a duct horseshoe vortex) is released from the trailing edge of the duct, and the circulation of the ith duct horseshoe vortex is denoted by Γ _Ei . The axial length of the duct horseshoe vortex may be about 5 times the diameter.

The coordinate system of the blade is shown in FIG. The number of blades B is arbitrary, but the first blade is set to coincide with the x-axis. The blade is represented by a lifting line, the lifting line is equally divided into N pieces, and the dividing width is dr. From the dividing point, the discharge vortices are spirally discharged according to the initial vortex surface moving velocity v ₀ ′ and the blade angular velocity Ω. The axial length of the discharge vortex may be about 5 times the diameter. The discharge vortices are discretized and approximated by broken lines as shown in FIG. Tip vortices coincide with the hypotenuses of the most fine grid of ducts.

As shown in FIG. 14, the control point is taken at the middle point of the divided lifting lines (hereinafter referred to as blade elements). A spiral horseshoe vortex consisting of the j-th shedding vortex, the j-th bound vortex, and the j + 1-th bound vortex is called the j-th horseshoe vortex, and the circulation of this horseshoe vortex is denoted by Γ _Bj .
3-2. Thrust and absorption power of fan

A velocity vector diagram of the flow entering the control point of the blade is shown in FIG. Let the y-direction component and the z-direction component of the induced velocity induced to the ith control point of the blade be v _Bi and w _Bi respectively.
w _Bi generally takes a negative value. Horizontal relative velocity U _Ti
Is given by the following equation.

同様に垂直方向の速度成分U_Pi（上向きが正）は

Similarly, the vertical velocity component U _Pi (upward is positive) is

で与えられる。流入角度φ_iは次式で与えられる。

Given by The inflow angle φ _i is given by the following equation.

(5)式で定義されるφ_iは一般に負の値をとる。i番目の翼素でのブレード取付角をθ_iとすると、有効迎角α_iは次式で与えられる。

The φ _i defined by the equation (5) generally takes a negative value. Assuming that the blade mounting angle at the _ith blade element is θ _i , the effective attack angle α _i is given by the following equation.

また流入する流れの速さV_iは

Also, the velocity V _i of the inflowing flow is

で与えられる。
i番目の翼素の循環はΓ_Biであるから、Kutta-Joukowski の定理より局所揚力dL_iは次式で与えられる。

Given by
Since the circulation of the i-th wing element is Γ _Bi , local lift force dL _i is given by the following equation according to Kutta-Joukowski's theorem.

ここでdrは翼素の分割幅である。ここで設計揚力係をC_L、未知数である翼弦長をc_iとすると、局所揚力dL_iは次式でも表される。

Here, dr is the division width of the wing element. Here, assuming that the design lift force is C _L and the unknown chord length is c _i , the local lift dL _i is also expressed by the following equation.

(8)式と(9)を等置して次式を得る。

Equations (8) and (9) are equal to obtain the following equation.

Accordingly, the Reynolds number Re is known from the chord length c _i and the flow velocity V _i , and C _D is determined from the selected airfoil and Reynolds number Re and the designed lift coefficient C _L. Local resistance dD _i using this C _D is given by the following equation.

これらd_Li、dD_iを図１６に示す。図１６より局所推力dT_iと局所水平分力dH_iはそれぞれ次式で与えられる。

These d _L i and dD _i are shown in FIG. Local Thrust dT _i and the local horizontal force dH _i than 16 are respectively given by the following equation.

(13)式より局所パワーdP_iは次式で表される。

From the equation (13), the local power dP _i is expressed by the following equation.

抵抗を考慮しない場合、局所推力dT_iはKutta-Joukowskiの定理より直ちに

If no resistance is taken into consideration, the local thrust dT _i is immediately determined by Kutta-Joukowski's theorem

で与えられる。同様に局所水平分力dH_iは

Given by Similarly, the local horizontal component dH _i is

で与えられる。負の符号がついているのはUP_iが通常負であるためである。ファンが発生する推力T_Fおよび吸収パワーPはそれぞれ次式で与えられる。

Given by The negative sign is because UP _i is usually negative. The thrust T _F generated by the fan and the absorbed power P are respectively given by the following equations.

ここでBはブレード枚数である。
３−３.ダクト推力の計算方法

Here, B is the number of blades.
3-3. Calculation method of duct thrust

If the fan diameter and the duct outlet diameters are the same, the ratio of the thrust T _F of the fan is generated from a simple momentum theory the total thrust is 50% thrust lip of the duct occurs (hereinafter duct thrust) T _D is the total thrust It is derived that the ratio to T is 50%.

The duct thrust T _D is obtained by multiplying the grid area by the negative pressure acting on the entire duct grid and summing the z-direction components. In practice, a grid with a normal normal does not generate a force in the z direction, so it is only necessary to calculate the duct grid that constitutes the flange and the lip of the duct. Since the control point roll point of the i-th duct grid of the flange and lip is between the inside with flow and the outside which is the water stop area, the velocity of the inside surface of the duct (u, v, w) ^T is _twice of (u _Dci , v _Dci , w _Dci ) ^T Also, the duct thrust dT ' _{Di at} the control point of the ith duct grid is from the nonstationary Bernoulli's theorem

となる。ここで右辺第一項はポテンシャルの時間微分の項であり、第２項は次式で与えられる。

It becomes. Here, the first term on the right side is a term of the time derivative of the potential, and the second term is given by the following equation.

Here, n _zi is the z component of the normal vector of the ith grid. When the local thrust dT ' _Di is summed over the duct, the term of the time derivative of the potential becomes zero due to the periodicity of the blade. Accordingly, the duct thrust T _D is given by the following equation.

３−４.ブレード上の誘導速度の計算
ファンの発生する推力T_F及び吸収するパワーPの計算には誘導速度を用いた。本節ではこれら誘導速度の計算式を導く。

3-4. Calculation of induction speed on blade The induction speed was used to calculate the thrust T _F generated by the fan and the power P absorbed. In this section we will derive formulas for these induction rates.

There are three types of vortices that cause the induced velocity, and there are N _B × B pieces of horseshoe vortices of the blade (circulation size Γ _Bi ), N _D vortex rings of the duct surface (circulation size Γ _Di ), duct horseshoe vortices There are N _E (the size of circulation Γ _Ei ).

First, the induced velocity (u _BBi , v _BBi , w _BBi ) ^T caused by all the horseshoe vortices of the blade at the control point of the ith blade is from Biot-Savart's law

And the influence coefficient can be expressed in the form (H. Harada, S. Kotake, Toshimasa Shiro: Optimal Design of Ducted Propellers, Proceedings of the Japan Aerospace Society of Japan, Vol. 59, 2011, pp. 298-305. ). Here, the subscripts i and j follow Einstein's rule. The notation of (22) is synonymous with the notation of the following formula

ダクト格子からの誘導速度、ダクト馬蹄渦からの誘導速度もそれぞれ

The induced velocity from the duct grid and the induced velocity from the duct horseshoe vortex respectively

および

and

と表記できる。以上がブレード上に引き起こされる誘導速度である。
３−５.ダクト表面上の境界条件
ブレードの馬蹄渦がダクト格子のコントロールポイントに引き起こす誘導速度の格子に対する法線成分u_DBを次式で表す。

It can be written as The above is the induced velocity induced on the blade.
3-5. Boundary Condition on Duct Surface The normal component u _DB to the lattice of the induced velocity which the horseshoe vortex of the blade causes at the control point of the duct lattice is expressed by the following equation.

The normal component u _DD to the lattice of the induced velocity induced by the duct lattice at the control point of the duct lattice, and the normal component u _DE to the lattice of the induced velocity induced by the duct horseshoe vortex at the control point of the duct lattice.

で表す。

Represented by

  The ducts are solid and there is no flow through the surface. Therefore, the normal component to the grid of the velocity induced to the control point of the duct grid should be zero. In other words

となる。この条件はダクト格子の数、つまりN_D個存在する。
３−６．ダクト後縁の境界条件

It becomes. This condition is the number of duct grids, that is, N _D.
3-6. Boundary condition of duct trailing edge

The figure which expanded the duct trailing-edge vicinity of FIG. 11 is shown in FIG. The ith duct horseshoe vortex is adjacent to the kth duct grid. The number of duct grids is N _D , of which only N _E are located at the trailing edge. N _E is much smaller number than N _D, appropriate vertical N _E columns, by a matrix K _K lateral N _D element row 1 or 0, the K _Kij gamma _Dj to the i-th duct trailing edge horseshoe vortex It can be an element of adjacent Γ _D.
The circulation behind the Kutta's condition must be zero. If this is expressed by an equation paying attention to the direction of circulation in FIG.

となる。
３−７．変数の削減

It becomes.
3-7. Variable reduction

The condition represented by the equation (29) is N _D which is the number of elements of Γ _D. In general, N _D will be tens of thousands. On the other hand, the number of elements N _B of Γ _B is as small as several tens to several hundreds. In this section, we eliminate Γ _D and reduce the amount of variables significantly.
By substituting the equations (26), (27) and (28) into the equation (29), the following equation is obtained.

(30)式より(31)式は

From equation (30), equation (31) is

となる。この式をさらに変形して

It becomes. Transform this equation further

を得る。ここで、

Get here,

とすると、次式を得る。

Then we get the following equation:

また(35)式を(30)式に代入して

Also, substituting equation (35) into equation (30)

を得る。
ブレードに引き起こされる誘導速度のx成分u_Bは(22)、(24)、(25)式より次式で表される。

Get
The x component u _B of the induction velocity induced by the blade is expressed by the following equation from the equations (22), (24) and (25).

この式に(35)式、(36)式を代入して

Substituting equations (35) and (36) into this equation

を得る。Γ_Bについてまとめると

Get When we summarize に つ い て_B

となる。ここで

It becomes. here

と定義すると、(39)式は

(39) is defined as

となる。v_B、w_Bにも同様の操作を行いY_B、Z_Bを定義し、

It becomes. v _B, the same operation to w _B Y _B, define a Z _B,

と表記する。
３−８．ダクト推力の二次形式での表現
本節では３−３で求めたダクト推力T_DをΓ_Bの二次形式で表す。

It is written as
3-8. Representation of Duct Thrust in Quadratic Form In this section, duct thrust T _D determined in 3-3 is represented in 二_B quadratic form.

  The velocity induced on the surface inside the duct is twice the velocity induced on the duct grid, and expressed by the following equation by the procedure similar to the derivation of the equations (41), (42) and (43) Can do.

ここでi番目のダクト格子の面積をS_i、法線ベクトルのz成分をn_Ziとし、

Let the area of the _ith duct grid be S _i and the z component of the normal vector be n _Zi

なる対角行列を用いれば、ダクト推力T_Dは次式で与えられる。

If the diagonal matrix is used, the duct thrust T _D is given by the following equation.

ここで

here

と定義することで(48)式は

By defining as (48)

と表記することが出来る。
３−９．抵抗を考慮しない場合の推力とパワー
抵抗を考慮しない場合、ファンの推力T_Fは(17)式に(15)式を代入し、さらに(3)式、(42)式を代入して次式で表される。

It can be written as
3-9. When not considering the thrust and the power resistance without considering the resistance, the fan thrust T _F substitutes the equation (15) into the equation (17), and further substitutes the equations (3) and (42) into the following equation Is represented by

または、ベクトルを用いて次式で表される。

Or it is represented by following Formula using a vector.

ここで A'は次式

Where A 'is

で与えられるN_B×N_Bの行列であり、b は

Is an N _B × N _B matrix given by, where b is

で与えられる要素数 N_Bの縦ベクトルである。
ファンの吸収パワーPは(18)式に(16)式を代入し、さらに(4)式，(43)式を代入して次式で表される。

This is a vertical vector of the number of elements N _B given by.
The absorption power P of the fan is expressed by the following equation by substituting the equation (16) into the equation (18) and further substituting the equations (4) and (43).

または、ベクトルを用いて次式で表される。

Or it is represented by following Formula using a vector.

ここでCは

Where C is

で与えられるN_B×N_Bの行列である。
ダクテッドファン全体の推力Tはファン推力T_Fとダクト推力T_Dの和であるから次式で表される。

It is an N _B × N _B matrix given by
The thrust T of the entire ducted fan is the sum of the fan thrust T _F and the duct thrust T _D , which is expressed by the following equation.

ここでAは

Where A is

で与えられるN_B×N_Bの行列である。
３−１０．最適化問題
静止推力を最大とするダクテッドファンの形状を決定する問題は、ここまでの計算より次の最適化問題を満たすΓ_Bを求める問題に帰着する。
minimize -T
subject to P -P₀ = 0
ここでP₀は設計者が設定する吸収パワーである。
Γ_Bが得られればブレード角θ_iは(6)式から、翼弦長c_iは(10)式より求められる。

It is an N _B × N _B matrix given by
3-10. Optimization problem The problem of determining the shape of the ducted fan that maximizes the static thrust results in the problem of finding Γ _B that satisfies the next optimization problem from the calculations so far.
minimize -T
subject to P-P ₀ = 0
Here, P ₀ is the absorption power set by the designer.
If Γ _B is obtained, the blade angle θ _i can be obtained from the equation (6), and the chord length c _i can be obtained from the equation (10).

When resistance is not considered, T of the objective function is given by equation (58), and P of constraint condition is given by equation (56). However, when the problem is optimized, the duct thrust T _D converges to a very large value. So as a constraint condition for limiting the values of T _D, the duct thrust T _D derived from simple momentum theory adds a condition that is equal to half of the total thrust T.
The improved problem has the following form.
minimize -T
subject to P-P ₀ = 0, TD = 0.5T

When resistance is taken into consideration, the thrust T can not be expressed in a quadratic form such as the equation (58), and the induction velocities v _B and w _B are determined from the equations (42) and (43). 12) The fan thrust T _F is determined from the equation up to the equation (17) and the equation (17), and from the duct thrust T _D determined from the equation (50)

として得られる。

Obtained as

Similarly, the absorption power P is determined from the equations (42) and (43), and the induction velocities v _B and w _B are determined from the equations (3) to (14) and the equation (18). Since the value of the drag coefficient is much smaller than the lift coefficient, the resistance solution does not differ significantly from the case where the drag is not considered.
3-11. Iterative calculation of optimization problems

  In order to solve this problem, as described in the previous chapter, it is necessary to repeat the calculation until the moving velocity of the spiral vortex converges. Here, in order to stabilize the repeated calculation, the vortex surface moving velocity is expressed by a quadratic function. The coefficients of this quadratic function are updated by curve fitting for each iteration.

Moreover, although T _D = 0.5 T required by the simple momentum theory is used as the constraint condition in 3-10, the duct thrust T _D is not necessarily 50% of the total thrust T even with a more realistic mathematical model. A parameter k (hereinafter referred to as a lip factor) is introduced to make T _D = kT, and the Trefftz surface thrust T _T and the total thrust T are made to coincide with the degree of freedom obtained by the introduction of this parameter. Specifically, when the Trefftz surface thrust T _T is larger than the total thrust T, k is decreased, and in the opposite case, k is increased to converge k.
3-12. Expansion to model considering hub

  As shown in FIG. 18, in reality, a hub for storing a motor or the like is required at the center of the duct. The hub is fixed at the center of the duct, houses the motor, and has a blade and spinner mounted on the motor shaft. Here, it models as follows. The spinner and the hub are the same diameter, the hub is a short cylinder, and the flow separates at the rear end of the hub and flows straight. Thus, the downstream end of the hub is a dead water region. The spinner and the hub play the role of the wingtip of the blade root as well as the ducting of the blade tip of the wingtip. Therefore, blade root vortices spirally exist on the spinner and hub surface. A lattice having a hierarchical structure is cut on the spinner surface and the hub surface as well as the duct surface, and the hub horseshoe vortex is emitted from the rear end of the hub like the duct horseshoe vortex. Also, the duct length is the length of the duct behind the blade rotation surface. The front of the duct is a portion forward of the blade surface of the straight portion of the duct.

Next we redefine the terms. Duct grid consists of flange, lip, duct front, duct, spinner, and hub surface grid. The duct horseshoe vortex consists of a horseshoe vortex emitted from the duct trailing edge and a horseshoe vortex emitted from the hub trailing edge. The duct thrust is the thrust generated by the lip, flange and spinner. In this way, if you redefine duct grids, duct horseshoe vortices, and duct thrusts, you can handle models that include spinners and hubs.
4. Calculation result

Here, we solved the optimization problem in consideration of resistance. In the solution, constraints were incorporated into the objective function by the penalty function method, and this objective function was minimized by simulated annealing. It converged to the same solution using various initial values. The calculation conditions are as follows.
<Calculation condition>
Duct diameter: 0.25 m
Spinner diameter: 0.06m
Absorbing power P ₀ : 1200 W
Number of blades B: 4
Duct length: 0.12 m
Duct front length: 0.015 m
Blade division number N _B : 80
Number of lattice layers n: 2
Duct grid number N _D : about 15,000 (without hub)
: About 30,000 (with hub)
Design lift coefficient C _L : 0.6
Shape resistance coefficient C _D : 0.014 (Re independent)
<Parameter>
Lip radius R _R : 0.01 to 0.06 m
Flange radius R _F : 0.25, 0.75 m

The Trefftz plane is sufficiently aft of the duct so it is not affected by the duct and is only affected by the blade discharge vortices and duct horseshoe vortices. Therefore, when calculating the Trefftz surface thrust T _T , a blade discharge vortex and a duct horseshoe vortex having a total length of 10 times the duct diameter were formed, and the Trefftz surface was taken at the center.

The shape of the blade when the flange radius R _F is 0.75 m obtained by optimization is shown on the top of FIG. 19 and the shape of the blade at 0.25 m is on the bottom of FIG. The case where the lip radius R _R is 0.01 m is shown by a solid line, and the case where it is 0.05 m is shown by an alternate long and short dash line. Similarly on the flange radius R _F obtained in the optimization when there is the hub of Figure 20 the shape of the time of 0.75 m, indicating the shape of the time of 0.25m below the figure 20. Here also, the case where the lip radius R _R is 0.01 m is indicated by a solid line, and the case of 0.05 m is indicated by an alternate long and short dash line. The vortex surface moving speed under the condition of the upper part of FIG. 20 is shown in FIG. The vortex surface moving speed shown in the figure is a value approximated by a quadratic function by curve fitting, and is the flow velocity used to determine the shape of the vortex surface at the time of optimization.

The thrust and lip factor k versus lip radius are shown in FIG. Hub flange radius R _F no is in ○ the value at 0.25 m, indicating the value at 0.75 □. Also, the value when the hub is present and the flange radius R _F is 0.25 m is indicated by Δ, and the value at 0.75 is indicated by ▽.
5. Consideration

As shown in FIGS. 19 and 20, a blade shape with an enlarged tip was obtained. This seems to be due to the fact that the duct wall inhibits the formation of the tip vortices and also uses the lip thrust. Also, the result is that the lip shape optimized for the lip radius is dependent. This is because the matrix T _{D of the} equation (50) which gives the duct thrust T _D depends on the shape of the duct, and the obtained result is expected to depend on the lip radius R _R.

The smaller the lip radius R _R, the larger the degree of expansion of the wing tip. This is considered to be because, as the lip radius is smaller, by increasing the axial flow velocity near the wall surface, it is possible to increase the duct thrust due to the flow velocity around the lip. In order to support this, the distribution of vortex surface movement velocity v ′ shown in FIG. 21 is nearly uniform at R _R = 0.05, and the velocity near the wall increases at R _R = 0.01. Further, as shown in FIG. 19, as the lip radius R _R is smaller, the thrust T is larger. This is because, as described above, the smaller the lip radius R _R , the larger the duct thrust. It can be seen from the graph of lip factor k (FIG. 22) that the smaller the lip radius R _{R, the} larger the duct thrust.

In simple momentum theory, the lip factor k is exactly 0.5, but in this calculation a value smaller than 0.5 was obtained. However, as seen in FIG. 22, the value of k is closer to 0.5 when the flange radius R _F is 0.75 m than when the flange radius R _F is 0.25 m, and the lip factor k approaches 0.5 when a larger flange is used. Seem.

When considering the hub, the blade shape is a shape in which the portion of the blade-shaped hub when the hub is not present is simply removed. However, when looking at the graph of thrust in FIG. 22, there is almost no decrease in thrust. On the contrary, the lip factor increases, and there is a tendency to use duct thrust more. In order to support this, as shown in FIG. 22, the condition in which the degree of enlargement of the tip is the largest is the case where there is a hub, the flange radius R _F is 0.75 m, and the lip radius R _R is 0.01 m. The

The duct thrust increases as the flange radius increases, so the thrust increases as the flange radius increases for the same fan diameter. However, when the flange radius _RF is calculated to be twice and six times the fan radius, the increase in thrust is about 2N, which is only about 3% of the total thrust 60N. Applications that use large flanges that add bulk and weight are believed to be limited to special cases.

In this method, optimization was carried out in consideration of the resistance, but in the Reynolds number region used in the calculation example, there was almost no difference between the optimum shape with and without the resistance taken into consideration. However, if the Reynolds number is extremely small and the drag coefficient can not be ignored, it is possible to obtain significantly different solutions.
6. Conclusion The design of ducted fan that maximizes the static thrust is expressed as an optimization problem and minimized.

The greatest feature of this calculation is that the thrust itself taking into consideration the effect of resistance is used as the objective function, which allows solutions to be uniquely determined regardless of the designer's ability, so optimization with a low Reynolds number can also be performed. The point and fine features are considered the duct thrust, the hub needed to make the blade, and the pitch of the wing tip discharge vortices which determine the shape of the discharge vortices by repeated calculation Using a hierarchically structured grid with half of the internal vortices, few variables with only a few dozens of optimizations in a short time, minimizing the optimization problem The obtained blade has a characteristic shape in which the tip is enlarged, and the degree of enlargement depends on the lip radius of the duct.
(Others)

The ducted fan according to the present invention can be used as a multicopter, a vertical take-off and landing (VTOL) machine, a CPU cooling fan, a radiator cooling fan, and the like. In addition, it can be used for tail fans of helicopters.
The present invention is not limited to the above-described embodiment, and can be variously modified and implemented, and the scope of the implementation is, of course, within the technical scope of the present invention.

DESCRIPTION OF SYMBOLS 1: Ducted fan 10: Duct 12: Exhaust port 14: Lip 15: Inner wall 20: Propeller 21: Hub 21 A: Wing tip 22: Bread 22: Blade 22 A: Wing tip 22 B: Near blade tip

The propeller 20 has a hub 21 arranged concentrically with the duct 10 and four blades 22 provided at equal intervals on the outer periphery of the hub 21. The number of blades 22 is typically four, but two to eight are preferable. Further, the number of blades 22 is, for example, four when the number of stators 50 is seven, so that the plurality of blades 22 and the number of common least of the number of stators 50 have large values. And the position of the stator 50 can be avoided at the same time, and noise due to interference can be reduced.
The hub 21 is rotationally driven by a motor 30 housed in a housing 40. Thereby, the blade 22 rotates.

Here, we solved the optimization problem in consideration of resistance. In the solution, constraints were incorporated into the objective function by the penalty function method, and this objective function was minimized by simulated annealing. It converged to the same solution using various initial values. The calculation conditions are as follows.
<Calculation condition>
Duct diameter: 0.25 m
Spinner diameter: 0.06m
Absorbing power P ₀ : 1200 W
Number of blades B: 4
Duct length: 0.12 m
Duct front length: 0.015 m
Blade division number N _B : 80
Number of lattice layers n: 2
Duct grid number N _D : about 15,000 (without hub)
: About 30,000 (with hub)
Design lift coefficient C _L : 0.6
Shape resistance coefficient C _D : 0.014 (Re independent)
<Parameter>
Lip radius R _R : 0.01 to 0.06 m
Flange radius R _F : 0.25, 0.75 m

Claims (13)

A duct with an R-shaped lip at the outlet,
And a propeller having a blade tip close to an inner wall of the duct, a chord length gradually decreasing toward the blade tip, and a portion extending from the vicinity of the blade tip to the blade tip or having a fixed length. Ducted fan. A ducted fan according to claim 1, wherein
Assuming that the chord length of the blade tip of the blade is CL _A and the chord length of the portion near the blade tip of the blade is CL _B :
1.0 × CL _B ≦ CL _A ≦ 1.3 × CL _B
That's a ducted fan. A ducted fan according to claim 1 or 2, wherein
The ducted fan, wherein the position near the blade tip where the chord length is enlarged or fixed is within 15% of the radius of the blade from the blade tip. A ducted fan according to any one of claims 1 to 3, which is:
The degree of gradual reduction of the chord length of the blade decreases as it approaches the blade tip. The ducted fan according to any one of claims 1 to 4, wherein
The ratio of the maximum chord length to the minimum chord length of the blade is 1.5 or more and 3.0 or less. The ducted fan according to any one of claims 1 to 5, wherein
When the inside diameter of the duct is R,
3/100 × R ≦ r ≦ 30/100 × R
A ducted fan having an R shape with a radius of curvature r. A ducted fan according to any one of claims 1 to 6, which is:
The ducted fan is longer than 25% of the inner diameter of the duct. The ducted fan according to any one of claims 1 to 7, wherein
The propeller has a hub for holding the blade,
The diameter of the hub is 10% or more and 50% or less of the inner diameter of the duct. A ducted fan according to claim 8, wherein
The hub holds two to eight of the blades.   A multicopter comprising the ducted fan according to any one of claims 1 to 9.   A vertical take-off and landing aircraft having the ducted fan according to any one of claims 1 to 9.   A CPU cooling fan comprising the ducted fan according to any one of claims 1 to 9.   A radiator cooling fan comprising the ducted fan according to any one of claims 1 to 9.

Images